Cayley lattices of finite Coxeter groups are bounded
type: Item
instance of: scholarly article
title: Cayley lattices of finite Coxeter groups are bounded
zbMATH Open document ID: 1097.06001
author: Nathalie Caspard
author: Claude Le Conte de Poly-Barbut
author: M. Morvan
published in: Advances in Applied Mathematics
publication date: 2004-10-12T00:00:00Z
review text: Given a poset \(P\) and an interval \(I\) in \(P\), an interval doubling \(P(I)\) replaces \(I\times\{0\leq1\}\) maintaining the order \(x< I\) by \(x< I\times\{0\}\) and \(x> I\) by \(x> I\times\{1\}\). If \(P\) can be obtained from \(\{0\leq 1\}\) by a finite sequence of interval doublings, then it is bounded. In this useful paper a class of lattices HH is introduced, probably of importance in its own right, whose elements are shown to be bounded. Coxeter lattices are obtained from the transitive closures of the Cayley graphs of finite Coxeter groups via the weak order for general Coxeter groups. They are shown to be members of HH, which makes HH an interesting class of lattices ipso facto and bounded. The relationship between construction and deconstruction as in HH can be observed on the Coxeter lattices with meaningful interpretations in terms of reflections and parabolic subgroups whose lattices may each be reached from the over-arching lattice by a series of interval contractions as shown. Due to the intimate connections between Coxeter groups and geometry, the notions and results introduced and obtained here also yield new interpretations via these connections for even greater interest as noted by the authors.
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 06B05
zbMATH DE Number: 2107240
zbMATH Keywords: Bounded lattice
zbMATH Keywords: Coxeter group
zbMATH Keywords: Doubling operation
zbMATH Keywords: Reflection
zbMATH Keywords: Semidistributivity
zbMATH Keywords: Zonotope
zbMATH Keywords: Coxeter lattice
reviewed by: J. Neggers
MaRDI profile type: Publication
full work available at URL: j.aam.2003.09.002
OpenAlex ID: W2081137745
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DOI: 10.1016/J.AAM.2003.09.002
description: scientific article; zbMATH DE number 2107240